2022/05/02 by Yamamoto, Yuto
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.00814
Let \ Zt \t be a one-parameter family of complex hypersurfaces of dimension d ≥ 1 in a toric variety. We compute asymptotics of period integrals for \ Zt \t by applying the method of Abouzaid--Ganatra--Iritani--Sheridan, which uses tropical geometry. As integrands, we consider Poincaré residues of meromorphic (d+1)-forms on the ambient toric variety, which have poles along the hypersurface Zt. The cycles over which we integrate them are spheres and tori which correspond to tropical (0, d)-cycles and (d, 0)-cycles on the tropicalization of \ Zt \t respectively. In the case of d=1, we explicitly write down the polarized logarithmic Hodge structure of Kato--Usui at the limit as a corollary. Throughout this article, we impose the assumption that the tropicalization is dual to a unimodular triangulation of the Newton polytope.